In the last post, I described bosonic string theory and its limitations. A natural method to augment the theory is to introduce supersymmetry, an established idea from quantum mechanics.
A Foray into Supersymmetry
It is imperative to take a momentary step back and describe supersymmetry in general, unrestricted to the confines of stringy physics. The premise is straightforward: supersymmetry posits the existence of one (or more!) particles corresponding to each existing particle in the model. The kicker is that this partner has the opposite “statistics”, meaning that a boson will be paired with a fermionic superpartner and vice versa. For such a radical proposal, it is somewhat puzzling that physicists place a great deal of faith in its existence!
Here’s why - supersymmetry automatically solves many pressing unresolved problems lurking in the Standard Model, including the small value of the Higgs boson relative to its theoretically expected value (it weighs in at a hefty 125 GeV, nonetheless). Furthermore, it modifies the behaviour of the fundamental forces (electromagnetic, weak and strong), causing them “align” at high energies and thus setting the stage for a grand unification, wherein all the forces are seen to be aspects of a single, unified, fundamental force, and their diverging characters emerge at low energies due to another Higgs mechanism. Now without supersymmetric corrections, the “couplings” of these forces come tantalisingly close, provoking mystery at this unnaturally small offset. This is probably the paramount empirical support of supersymmetry, the other compelling feature propelling it beyond its adversaries being that it makes no changes to ordinary energy physics, and so is immediately compatible with the Standard Model and the substantial experimental checks it has been subjected to. Among other things, this unification predicts that the proton is unstable, and can decay to two lighter particles: an antielectron and a pion. Finally, certain attractive supersymmetric models provide clear-cut candidates for dark matter - the lightest supersymmetric partners of neutrinos.
Experimental observations have not treated this conjectured symmetry lightly. While the LHC could validate the existence of the Higgs boson, it could not find a trace of superparticles, to much disbelief. Proton decay experiments have been conducted for years with tonnes of heavy water in salt mines, but have yielded negative results. Lest one cast premature aspersions on the theory, let me quickly add that these results present no significant blow to supersymmetry - the supersymmetric scale could simply be higher than the energies probed by the LHC.
Least troubled by the news were the mathematically-inclined theoreticians - SUSY is a prized member of many a toolkit. With seemingly more chaos with the introduction of more particles, it may be surprising that supersymmetric theories are often simpler than the base model! This is because SUSY constrains the theory, placing restrictions on its freedom to foster complexity. Take quantum chromodynamics, a consumer of millions of supercomputer hours a year. Rigorous, mathematically sound descriptions of the mass gap, confinement, and strongly coupled behaviour at low energy are thorns in the otherwise beautiful, “non-abelian gauge” theory. If anyone proclaims otherwise, they are kindly advised to claim their $1 million from the Clay Mathematics Institute. But when supersymmetrised, the resulting SQCD suddenly becomes amenable to mathematical torture and inspection, and what’s more, we can probe the strong coupling effects exactly! As a crazier example, consider “$\mathcal N=4$ Super Yang Mills” theory - this is the most symmetric theory without gravity in 4 dimensions, with each particle having four superpartners. The Lagrangian is as cool as the name:
Incredible though it may seem, this theory is completely conformal, or scale-invariant, meaning that it looks the same zoomed in or out (were you paying attention in the previous blog post?) and as such, it is often used a toy model, especially in higher-dimensional physics.
The Superstring
Speaking of higher dimensional physics, this detour has lit the path for resolving the fermion problem in string theory: we ought to pepper in supersymmetry to get a consistent, all-encompassing, finite, quantum-gravitational theory. Since this is easier said than done, revisiting our dear friend the bosonic string offers us a moment of solace. Though I did not mention this very explicitly in the previous post, there are two clearly different types of strings: open strings, with free ends, and closed strings, whose ends are joined to form a loop. Now a theory with open strings is necessarily a theory with closed strings, because an open string can move in a circle to trace out a cylinder. But this is just the worldsheet for a closed string propagating! Bear in mind that the converse does not hold: there is no mechanism for a closed string to snap into an open one, and so there can exist theories of only closed strings.
The analysis of the closed string spectrum is roughly equivalent to analysing two copies of the open string, combined together. This can be thought of as vibrations on a looped string being a combination of left-moving waves and right-moving waves, at least classically. The coordinate on the string is naturally intepreted as a bosonic field and the usual quantization procedure (using the dictionary from ordinary quantum mechanics to translate a classical theory into its quantum counterpart) yields bosonic string theory, described in the previous post.
However we return to the battle, war-hardened and bruised with fermionic deficiency, this time brandishing supersymmetry (perhaps a little too fiercely, could you tone it down a notch?). And this dictates that we add a fermionic coordinate as a superpartner to the bosonic coordinate, for both the left-moving and right-moving modes. We simply analyse each set of modes separately, eventually combine them together and several pages of mathematical toil later, we’ve done it. We’ve constructed superstring theory (or rather, Green and Schwarz did, 30 years ago).
Or have we? There are really two pictures associated to string theory, one of string floating around, interacting in a high-dimensional spacetime, and one defined by two-dimensional interacting quantum field theories on donut-like surfaces forming the string worldsheet. Ontologically, string theory does not make a distinction between the two. But the supersymmetric supplement that we added above is only for the worldsheet theory. The string spectrum is not supersymmetric in spacetime. But if we want string theory to be an accurate description of reality, it should have spacetime supersymmetry to reduce to the field-theoretic picture described previously! A deeper analysis suggests more gripping problems: there is still a tachyon in the theory, and the presence of a massless gravitino in the theory (the superpartner of the graviton) leads to a breakdown of causality and some sacred mathematical tenets of the theory if spacetime supersymmetry is not present.
At this stage it seems string theory is well and truly done for. But the emotional rollercoaster continues. Gliozzi, Scherk and Olive discovered that in order to upgrade this currently non-physical theory to a consistent one, one must truncate the space of states by enacting the GSO projection. Amazingly, this solves every single one of the problems presented thus far. The tachyon does not survive the projection; it is removed from the theory. The number of degrees of freedom at each mass level match between the bosonic and fermionic sector, the coveted spacetime supersymmetry has been obtained, and causality and “gauge invariance” are restored. Superstring theory has finally come to life.
But wait (oh no…). Recall how the closed string is composed of left-moving and right-moving parts? Well, I conveniently forgot to mention that there are two inequivalent choices of GSO projection, and we are free to choose either the same or different ones for each direction. This means we have constructed not one, but two superstring theories!
But why? What’s the difference? Well that’s a story for another blog post.